SMP #8 Look for & Express Regularity in Reasoning
The North Carolina Unpacked Standards provide a summary statement about the SMPs for each grade level:
Kindergarten: Mathematically proficient students in Kindergarten begin to look for regularity in problem structures when solving mathematical tasks. Likewise, students begin composing and decomposing numbers in different ways. For example, in the task “There are 8 crayons in the box. Some are red and some are blue. How many of each could there be?” Kindergarten students are expected to realize that the 8 crayons could include 4 of each color (4+4 = 8), 5 of one color and 3 of another (5+3 = 8), etc. For each solution, students repeated engage in the process of finding two numbers that can be joined to equal 8.
1st Grade: Mathematically proficient students in Grade 1` begin to look for regularity in problem structures when solving mathematical tasks. For example, when adding up three one-digit numbers and using the make 10 strategy or doubles strategy, students engage in future tasks looking for opportunities to employ those same strategies. For example, when solving 8+7+2, a student may say, “I know that 8 and 2 equal 10 and then I add 7 to get to 17. It helps to see if I can make a 10 out of 2 numbers when I start.” Further, students use repeated reasoning while solving a task with multiple correct answers. For example, in the task “There are 12 crayons in the box. Some are red and some are blue. How many of each could there be?” Grade 1 students are expected to realize that the 12 crayons could include 6 of each color (6+6 = 12), 7 of one color and 5 of another (7+5 = 12), etc. In essence, students are repeatedly finding numbers that will add up to 12.
2nd Grade: Mathematically proficient students in Grade 2 begin to look for regularity in problem structures when solving mathematical tasks. For example, after solving two digit addition problems by decomposing numbers by place (33+ 25 = 30 + 20 + 3 +5), students may begin to generalize and frequently apply that strategy independently on future tasks. Further, students begin to look for strategies to be more efficient in computations, including doubles strategies and making a ten. Lastly, while solving all tasks, Grade 2 students accurately check for the reasonableness of their solutions during, and after completing the task.
3rd Grade: Mathematically proficient students in Grade 3 notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking themselves, “Does this make sense?”
4th Grade: Mathematically proficient students in Grade 3 notice repetitive actions in computation and look for more shortcut methods. For example, students may use the distributive property as a strategy for using products they know to solve products that they don’t know. For example, if students are asked to find the product of 7 x 8, they might decompose 7 into 5 and 2 and then multiply 5 x 8 and 2 x 8 to arrive at 40 + 16 or 56. In addition, third graders continually evaluate their work by asking themselves, “Does this make sense?”
5th Grade: Mathematically proficient students in Grade 5 use repeated reasoning to understand algorithms and make generalizations about patterns. Students connect place value and their prior work with operations to understand algorithms to fluently multiply multi-digit numbers and perform all operations with decimals to hundredths. Students explore operations with fractions with visual models and begin to formulate generalizations.
Thinkmath.edc.org has a discussion about SMP#8 as does Wisconsin.
Illlustrative Mathematics has video examples of SMP #8 from 2nd, 4th, and 5th grades.
See also the video example below.
Noristown Unified School District in Pennsylvania has published some outstanding newsletters which focus on each of the Standards for Mathematical Practice. SMP Newsletter #8 can be found here.
